Polynomial Equations & Inequalities

Sketch and analyze graphs of polynomial functions. Math 30-1 Alberta.

Lesson 2.5 of Polynomial Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction to Solving Equations. Learn what it means to solve an equation, understand the Zero Product Property, and discover the general strategy for solving any polynomial equation.
• Solving Linear Equations. Review the fundamentals of solving linear equations (degree 1) using algebraic techniques.
• Solving Quadratic Equations. Master three powerful methods for solving quadratic equations: factoring, quadratic formula, and completing the square.
• Solving Higher-Degree Polynomial Equations. Apply factoring theorems and division techniques to solve cubic, quartic, and higher-degree polynomial equations.

Inside the lesson — a free preview

What Does It Mean to "Solve" an Equation?

To solve an equation means to find all values of the variable that make the equation true. These values are called solutions or roots.

Example:. For the equation x + 5 = 8:

Types of Polynomial Equations

Polynomial equations are classified by their degree:

Why Learn Multiple Solution Methods?

Different equations require different approaches:

Method toolbox:.

Remember:. The more tools you have, the more equations you can solve!

The Zero Product Property

This is the most important property for solving polynomial equations:

Zero Product Property:. If a × b = 0, then a = 0 OR b = 0 (or both)

Why this matters:. Once we factor a polynomial equation, we can set each factor equal to zero and solve.

Example:. (x - 2)(x + 5) = 0

By Zero Product Property:

Solutions:. x = 2 or x = -5

The General Strategy for Solving Polynomial Equations

5-Step Process:.

Key insight:. This strategy works for ANY polynomial equation that can be factored!

Overview of This Lesson

What you'll learn in this lesson:.

Let's begin!

What is a Linear Equation?

A linear equation is an equation that can be written in the form:

Standard form:. ax + b = 0

where:

Characteristics:.

The 5-Step Process for Solving Linear Equations

Step 1: Eliminate fractions (if present). Multiply entire equation by the LCD (Lowest Common Denominator)

Step 2: Eliminate parentheses and combine like terms. Use distributive property and simplify

Step 3: Gather variable terms on one side. Get all x terms on one side, constants on the other

Step 4: Divide by the coefficient. Isolate x by dividing both sides

Step 5: Verify your answer. Substitute back into original equation

Worked example: Detailed Example 1: Linear Equation with Fractions

Question. Solve: (x)/(3) + 1 = (2x - 2)/(7)

Step 1: Eliminate fractions. Find the LCD of 3 and 7:

LCD = 21

Multiply EVERY term by 21:

21 · (x)/(3) + 21 · 1 = 21 · (2x - 2)/(7)

Simplify each term:

New equation (no fractions):. 7x + 21 = 6x - 6

Step 2: Eliminate parentheses and combine like terms. Already done! We have:

7x + 21 = 6x - 6

Step 3: Gather variable terms on one side. Subtract 6x from both sides:

7x - 6x + 21 = 6x - 6x - 6

x + 21 = -6

Subtract 21 from both sides:

x + 21 - 21 = -6 - 21

x = -27

Step 4: Divide by coefficient. The coefficient of x is already 1, so we're done.

x = -27

Step 5: Verify. Substitute x = -27 into the original equation:

✅ Solution:. x = -27

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